We now recall the (\(q\) -)discrete analogue of
\(D_z\) -residues introduced in [1]}, [2]}.
Let \(\phi \)  be an automorphism of \(F(z)\)  that fixes \(F\) . For a polynomial \(p\in F[z]\) ,
we call the set \(\lbrace \phi ^i(p)\mid i\in  {\mathbb {Z}}\rbrace \)  the \(\phi \) -orbit of \(p\) , denoted by \([p]_{\phi }\) .
Two polynomials \(p, q\in F[z]\)  are said to be \(\phi \) -equivalent (denoted as \(p\sim _{\phi } q\) ) if they are in the same \(\phi \) -orbit, i.e., \(p = \phi ^i(q)\)  for some \(i\in  {\mathbb {Z}}\) .
For any \(a, b \in F(z)\)  and \(m\in  {\mathbb {Z}}\) , we have
\(\frac{a}{\phi ^m(b)} = \phi (g) - g + \frac{\phi ^{-m}(a)}{b},\) 
